FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau GASCOM 2016 03 juin 2016 Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Outline of the talk 1 Motivations 2 321-avoiding permutations and alternating diagrams 3 Periodic parallelogram polyominoes 4 Marked heaps of segments 5 Computations of the corresponding generating functions Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Reduced decompositions The symmetric group S n is generated as a Coxeter group by the set S of simple transpositions s i = ( i , i + 1) with s 2 = 1 Relations: s i s i +1 s i = s i +1 s i s i +1 (braid relations) s i s j = s j s i if j � = i ± 1 (commutation relations) Definition (Length) ℓ ( w ) = minimal l such that w = s 1 s 2 · · · s l with s i ∈ S Such a minimal word is a reduced decomposition of w . Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform one into the other. Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Reduced decompositions The symmetric group S n is generated as a Coxeter group by the set S of simple transpositions s i = ( i , i + 1) with s 2 = 1 Relations: s i s i +1 s i = s i +1 s i s i +1 (braid relations) s i s j = s j s i if j � = i ± 1 (commutation relations) Definition (Length) ℓ ( w ) = minimal l such that w = s 1 s 2 · · · s l with s i ∈ S Such a minimal word is a reduced decomposition of w . Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform one into the other. Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Reduced decompositions The symmetric group S n is generated as a Coxeter group by the set S of simple transpositions s i = ( i , i + 1) with s 2 = 1 Relations: s i s i +1 s i = s i +1 s i s i +1 (braid relations) s i s j = s j s i if j � = i ± 1 (commutation relations) Definition (Length) ℓ ( w ) = minimal l such that w = s 1 s 2 · · · s l with s i ∈ S Such a minimal word is a reduced decomposition of w . Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform one into the other. Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Fully commutative elements Definition An element w is fully commutative if given two reduced decompositions of w , there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red( w ) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. C 2 C 3 C 1 C 4 Red( w ) = Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Fully commutative elements Definition An element w is fully commutative if given two reduced decompositions of w , there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red( w ) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. C 2 C 3 C 1 C 4 Red( w ) = Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Fully commutative elements Definition An element w is fully commutative if given two reduced decompositions of w , there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red( w ) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. C 1 Red( w ) = then w is FC Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Previous works [Billey–Jockush–Stanley (1993)] 321-avoiding permutations in S n correspond to fully commutative elements of type A n − 1 [Barcucci–Del Lungo–Pergola–Pinzani (2001)] enumeration of S n (321) with respect to the inversions: nice expression. [Green (2001)] 321-avoiding affine permutations in � S n correspond to fully commutative elements of type � A n − 1 . [Hanusa–Jones (2010)] enumeration of � S n (321) with respect to the inversions (or Coxeter length): nice periodicity properties but very complicated expression for the GF. [Biagioli–Jouhet–Nadeau (2014)] enumeration of S n (321) and � S n (321) with respect to the inversions using alternating diagrams and Motzkin-type lattice walks: recursive GF. Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Enumeration of FC elements by length Question Enumerate FC elements by length for any finite or affine Coxeter group W ? Compute the generating series � � q ℓ ( w ) and W FC ( q ) = W FC ( q , x ) = W FC ( q ) x n w ∈ FC n ≥ 0 Theorem (B., Jouhet, Nadeau, 2014) We computed W FC ( q ) for any finite and affine W . If W is affine, the coefficients of W FC ( q ) form an ultimately periodic sequence. The main tool is to encode FC elements by certain lattice paths. BUT this does not give rise to nice expressions for W FC ( q , x ). Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions Enumeration of FC elements by length Question Enumerate FC elements by length for any finite or affine Coxeter group W ? Compute the generating series � � q ℓ ( w ) and W FC ( q ) = W FC ( q , x ) = W FC ( q ) x n w ∈ FC n ≥ 0 Theorem (B., Jouhet, Nadeau, 2014) We computed W FC ( q ) for any finite and affine W . If W is affine, the coefficients of W FC ( q ) form an ultimately periodic sequence. The main tool is to encode FC elements by certain lattice paths. BUT this does not give rise to nice expressions for W FC ( q , x ). Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
FC elements 321-avoiding elements Parellolograms polyominoes Generating functions An example Generating functions � A FC n − 1 ( q ): the first ones are 2 ( q ) = 1 + 3 q + 6q 2 + 6q 3 + 6q 4 + · · · � A FC 3 ( q ) = 1 + 4 q + 10 q 2 + 16q 3 + 18q 4 + 16q 5 + 18q 6 + · · · � A FC 4 ( q ) = 1 + 5 q + 15 q 2 + 30 q 3 + 45 q 4 � A FC + 50q 5 + 50q 6 + 50q 7 + 50q 8 + 50q 9 + · · · 5 ( q ) = 1 + 6 q + 21 q 2 + 50 q 3 + 90 q 4 + 126 q 5 + 146 q 6 � A FC + 150q 7 + 156q 8 + 152q 9 + 156q 10 + 150q 11 + 158q 12 + 150q 13 + 156q 14 + 152q 15 + 156q 16 + 150q 17 + 158q 18 + · · · The coefficients of � A FC n − 1 ( q ) are ultimately periodic of period dividing n , and the periodicity starts from degree 1 + ⌈ ( n − 1) / 2 ⌉⌊ ( n + 1) / 2 ⌋ . Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations
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